searching the database
Your data matches 53 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000931
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 1
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001172
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,3,6,7,4,8] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,2,5,3,6,8,4,7] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,4,5,8,7] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,2,7,8,4,5,6] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,4,2,6,8,5,7] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,5,6,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,2,8,5,7] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,3,6,7,2,4,5,8] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,4,2,3,6,8,5,7] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3,7,8] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,2,5,6,3,8,7] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,7,8,5] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,2,8,3,6] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,4,6,2,3,7,5,8] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ?
=> ?
=> ? = 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,5,6,2,3,7,4,8] => ?
=> ?
=> ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ?
=> ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ?
=> ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ?
=> ? = 0
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001066
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [7,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [6,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [6,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [7,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4]
=> ?
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4]
=> ?
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,3]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,3]
=> ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3]
=> ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 0 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000052
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,4,7,5,6,8] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,5,8,6] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3,5,4,7,8,6] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,6,7] => [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,3,5,6,4,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,3,5,6,7,4,8] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,3,5,7,4,6,8] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,4,8,6] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,7,8,4,6] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,4,6,7] => [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,4,5,7,8] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,6,4,5,8,7] => [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5,8] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,4,7,8,5] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,4,8,5,7] => [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,4,5,8] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,4,5] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,4,5,7] => [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,4,5,6,8] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,4,5,8,6] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,4,8,5,6] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,4,5,6,7] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,4,3,5,7,8,6] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,2,4,3,6,7,5,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,4,3,6,7,8,5] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,3,6,8,5,7] => [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,7,5,6,8] => [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,5,8,6] => [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,2,4,3,7,8,5,6] => [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,5,6,7] => [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,2,4,5,3,7,6,8] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St001483
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
St001483: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000118
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[[.,.],.]],[.,.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],[[.,.],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[[[.,.],.],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[.,.],[.,.]],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,[.,[.,.]]],.]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[[.,.],[.,.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[[[.,.],.],[.,.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,[.,.]],[.,.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[.,[[.,.],.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,.],[[.,.],.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[[.,[.,.]],.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,.],[.,[.,.]]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[[.,.],[.,.]]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[[.,.],.]]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[.,[[.,.],.]],.],[.,.]],.],.]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> ? = 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St001167
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,2,1,2] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,2,3] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,3,2] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,3,2] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,2,1,2] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,2,2,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,2,3] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,3,2] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,3,2] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St000366
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [4,3,2,1,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [3,2,1,5,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,5,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,5,4,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [3,2,1,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,5,4,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,5,4,6,7] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [3,2,1,5,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,5,4,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,5,4,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,5,4,3,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,4,6,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [4,3,2,1,6,5,7] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,4,6,5,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,4,6,5,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [4,3,2,1,6,5,7] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,6,5,4,7] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,6,5,4,7] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,6,5,4,7] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [2,1,3,4,5,7,6] => ? = 0
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [3,2,1,4,5,7,6] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [2,1,4,3,5,7,6] => ? = 0
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [2,1,4,3,5,7,6] => ? = 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000371
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [4,3,2,1,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [3,2,1,5,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,5,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,5,4,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [3,2,1,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,5,4,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,3,5,4,6,7] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [3,2,1,5,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,5,4,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,5,4,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [2,1,5,4,3,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,4,6,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [4,3,2,1,6,5,7] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,4,6,5,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,4,6,5,7] => ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,4,6,5,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,4,3,6,5,7] => ? = 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [4,3,2,1,6,5,7] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,6,5,4,7] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,6,5,4,7] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,5,4,7] => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> [3,2,1,6,5,4,7] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [1,6,5,4,3,2,7] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [2,1,3,4,5,7,6] => ? = 0
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [3,2,1,4,5,7,6] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [2,1,4,3,5,7,6] => ? = 0
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [2,1,4,3,5,7,6] => ? = 0
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000731The number of double exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000932The number of occurrences of the pattern UDU in a Dyck path. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001330The hat guessing number of a graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000444The length of the maximal rise of a Dyck path. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000982The length of the longest constant subword. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001948The number of augmented double ascents of a permutation. St000392The length of the longest run of ones in a binary word.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!